Vegetarianism is the practice of abstaining from the consumption of meat (red meat, poultry, seafood, and the flesh of any other animal), and may also include abstention from by-products of animal slaughter. Vegetarianism may be adopted for various reasons.
It is not necessary to take it as Najis unless you are sure that food was touched with wetness by a Najis hands. Food cooked with out been touched or by wearing gloves is not Najis.
This would depend on a few things:
1. The fatwa of your Marja' taqleed, regarding Ahlul Kitab being tahir or not.
2. The non-Muslim being from Ahlul Kitab, or not.
3. Whether they are wearing gloves, or if there is any cross-contamination.
In these mentioned cases, cross-contamination with najasah would make the food najis as well, and therefore haram to eat.
Unless you know that the common process in these restaurants is that they wash the pizza pan, change gloves, wash the knife, and so on. You can ask them these questions, and you can ask that the pizza not be cut, to avoid contamination from najasah.
As for using the same oven, that would not be a problem, because the pizza is put on a pan and then in the oven. If a pan is not used, then this is when there would be a problem.
In regards to french fries, or "chips", not only must the oil itself be halal, it must not have been used for cooking anything non-halal that is najis.
The oil must not be animal based, so it should be vegetable oil. What is cooked in it must not be najis, like meat that has not been Islamically slaughtered. However, if it is not halal to eat, but tahir, then there would not be a problem, like non-halal sea food. This is something you will experience in a fish and chips shop, where they might cook fish that does not have scales, or maybe even crab, in the same oil as the chips, or halal fish. In this case there is no problem, as long as there are no pieces of that food sticking to the food that you are going to eat.
And Allah knows best.